Gates
Logic gates are the basic elements that make up a
digital system. The electronic gate is a circuit that is able to operate
on a number of binary inputs in order to perform a particular logical
function. The types of gates available are the NOT, AND, OR, NAND, NOR,
exclusive-OR, and the exclusive-NOR . The gate is a digital circuit with one or more input
voltages but only one output voltage. The operation of a logic gate can
be easily understood with the help of Truth Table. A truth table is a
table that shows all the input-output possibilities of a logic circuit.
AND GATE: The AND gate produces the AND logic function that is the output is 1 if the input A and B are both equal to 1, other wise the output will be zero...In other words we can say that it is the product of both the inputs to a single output..
NOT: The inverter performs a basic logic function called inversion or complementation. The inverter changes one logic level to its opposite level. when a HIGH level is applied to an inverter input, a LOW level appear on its output. When LOW level is applied to it input, a HIGH level appear on its output.
OR: The OR Gate has two or more inputs ans one output. The operations of the OR gate is to determine when one or more of its inputs are HIGH and to produce a HIGH on its output to indicate this condition. In other words that we can say that it is the addition of both the inputs ot more inputs....
NAND: The term NAND is a contraction of NOT-AND and implies and AND function with a complemented (inverted) output. The NAND gate is a universal gate as it can be used to construct an AND gate, an OR gate an inverter or any combination of these functions. The logical operation of the NAND gate is such that a LOW output occurs only when all inputs are HIGH. When any of the inputs is LOW, the output will be HIGH.
NOR:The term NOR Gate is the contraction of NOT-OR gate is a universal gate. The logic operation of the NOR gate is that a LOW output occurs when any if its inputs is high. Only when all of its inputs are LOW, the output is HIGH.
ExOR: The Ex-OR gate is an abbreviation for Exclusive-OR gate. An Ex-OR gate has two or more inputs and one output. The Logic operation of the Ex-OR gate is the output is HIGH only when the odd number of its input. In other words if both are input are same output will always low and both the input are different the output will always be high..
ExNOR:The EX-NOR gate is the contraction of NOR-Exclusive-OR gate.An Ex-Nor gate has two or more inputs and has only one output. The logic operation of the EX-NOR gate is the output is HIGH only when the even number of its inputs or inputs having all zeros.
AND GATE: The AND gate produces the AND logic function that is the output is 1 if the input A and B are both equal to 1, other wise the output will be zero...In other words we can say that it is the product of both the inputs to a single output..
NOT: The inverter performs a basic logic function called inversion or complementation. The inverter changes one logic level to its opposite level. when a HIGH level is applied to an inverter input, a LOW level appear on its output. When LOW level is applied to it input, a HIGH level appear on its output.
OR: The OR Gate has two or more inputs ans one output. The operations of the OR gate is to determine when one or more of its inputs are HIGH and to produce a HIGH on its output to indicate this condition. In other words that we can say that it is the addition of both the inputs ot more inputs....
NAND: The term NAND is a contraction of NOT-AND and implies and AND function with a complemented (inverted) output. The NAND gate is a universal gate as it can be used to construct an AND gate, an OR gate an inverter or any combination of these functions. The logical operation of the NAND gate is such that a LOW output occurs only when all inputs are HIGH. When any of the inputs is LOW, the output will be HIGH.
NOR:The term NOR Gate is the contraction of NOT-OR gate is a universal gate. The logic operation of the NOR gate is that a LOW output occurs when any if its inputs is high. Only when all of its inputs are LOW, the output is HIGH.
ExOR: The Ex-OR gate is an abbreviation for Exclusive-OR gate. An Ex-OR gate has two or more inputs and one output. The Logic operation of the Ex-OR gate is the output is HIGH only when the odd number of its input. In other words if both are input are same output will always low and both the input are different the output will always be high..
ExNOR:The EX-NOR gate is the contraction of NOR-Exclusive-OR gate.An Ex-Nor gate has two or more inputs and has only one output. The logic operation of the EX-NOR gate is the output is HIGH only when the even number of its inputs or inputs having all zeros.
Boolean algebra...
Boolean algebra is a form of mathematics developed by English mathematician George Boole (1815–1864). Boole created a system by which certain logical statements can be expressed in mathematical terms. It deals in binary variable and logic operation. The letter are denoted by by letter A,B,x,y..The three basic logic operation are AND, OR and complement.. A Boolean function can be expressed algebraically with binary variable, the operation symbol and equal sign...For a given value of variable the boolean function can be either 1 or 0.... Like an example F=x+ y'z......Here f is the function which is equal to 1 if x is 1 .....If the value of x is zero it fully depends on the variable y and z.....
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BASIC RULES IN BOOLEAN:
1. x+0=x
2. x=1=1
3. x+x=x
4. x+x'=1
5.x+y=y+x
6.x+(y+z)=(x+)+x
7.x(y+z)=xy+yz
8.(x+y)'=x'y'
9.(x')'=x
10. x.0=0
11. x.1=x
12.x.x=x
13. x.x'=0.....
DE Morgan theorem.......
De Morgan theorem is very important in dealing in NAND and NOR gates...It states that A NOR gate that perform (x+y)' is equal to the function x'y'....And a NAND function cab be written as (xy)' is equal to x'+y'.....The following circuit diagram is shown in the figure which can easily show us the equality of the variables....NOTE: draw a diagram and draw a table of your own showing operation..
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K MAP......
A Karnaugh map (K-map for short) is a useful tool used in the simplification of combinational boolean equations and the creation of sequential logic circuits.The size of a Karnaugh map can be very large, however a size of four columns by four rows is easier to understand than any larger maps... Using a K-map, expressions with two to four variables are easily minimized. Expressions with five to six variables are more difficult but achievable, and expressions with seven or more variables are extremely difficult (if not impossible) to minimize using a K-map....A maxterm is a Boolean expression resulting in a 0 for the output of a single cell expression, and 1s for all other cells in the Karnaugh map, or truth table. A minterm is a Boolean expression resulting in 1 for the output of a single cell, and 0s for all other cells in a Karnaugh map, or truth table.
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The following diagram shows the kmap of 2 by 2....
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This shows the 4 by 4 kmap example....
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SUM OF PRODUCT: A minimum sum-of-products (SOP) expression for a function is defined as a sum of products which has A minimum number of terms , A minimum number of literals of all those expressions
which have the same minimum number of terms....
PRODUCT OF SUM: A minimum product-of-sums (POS) expression is defined as a POS which has A minimum number of factors, A minimum number of literals of all those expressions which have the same minimum number of factors...
which have the same minimum number of terms....
PRODUCT OF SUM: A minimum product-of-sums (POS) expression is defined as a POS which has A minimum number of factors, A minimum number of literals of all those expressions which have the same minimum number of factors...